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By Keith M. Ball, Vitali Milman
Convex our bodies are straight away uncomplicated and amazingly wealthy in constitution. whereas the classical effects return many many years, in past times ten years the vital geometry of convex our bodies has gone through a dramatic revitalization, led to by means of the creation of equipment, effects and, most significantly, new viewpoints, from likelihood concept, harmonic research and the geometry of finite-dimensional normed areas. This assortment arises from an MSRI software held within the Spring of 1996, related to researchers in classical convex geometry, geometric sensible research, computational geometry, and comparable components of harmonic research. it really is consultant of the easiest study in a truly lively box that brings jointly principles from a number of significant strands in arithmetic.
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Additional info for Convex geometric analysis
F ∈ Pa if 0 ≤ f < a is convex. f ∈ Pa ⇒ erτ Sτ f ∈ Pa . We are now going to introduce slightly larger classes of payoff functions than the classes Pa , a > 0. To this end, let a > 0 be given and suppose f : R m + → [0, a[ is a continuous function and set for fixed τ > 0, g = gτ = − 1 √ ln(1 − f /a) . σm τ (23) GEOMETRIC INEQUALITIES IN OPTION PRICING 49 We shall say that the the function f belongs to the class Pa,m if the function Φ−1 (P[gτ (x1 eσ1 √ τ c1 ,G , . . , xm eσm √ τ cm ,G ) ≤ s]) − s, s>0 is non-decreasing for every x ∈ R m + and τ > 0.
In fact, already the early paper  by Merton treats a variety of convexity properties of puts and calls, sometimes without any distributional assumptions on the underlying stock prices. Here, however, it will always be assumed that the price process X(t) = (X1 (t), . . , Xm (t)), t ≥ 0, of the underlying risky assets X1 , . . , Xm is governed by a so called joint geometric Brownian motion. Furthermore, all options will be of European type and so, from now on, option will always mean option of European type.
Then f ∈ C if and only if f > 0 and f (xeξ ) ≤ f (x)e ξ ∞ m x ∈ Rm +, ξ ∈ R . , (12) Moreover , f ∈ Ca if and only if a + f (xeξ ) ≤ (a + f (x))e ξ ∞ m x ∈ Rm +, ξ ∈ R . , In particular , any f ∈ Ca is a payoff function. Proof. Suppose first that f > 0 and set g(ξ) = ln f (eξ ), ξ ∈ Rm. Clearly, the inequality (12) just means that g ∈ Lip∞ (R m ; 1). Now let f ∈ C. e. e. and, hence, g ∈ Lip∞ (R m ; 1). Conversely, suppose g ∈ Lip∞ (R m ; 1). Then f ∈ Liploc (R m + ) and (13) holds. Accordingly, the inequality (11) must be true.
Convex geometric analysis by Keith M. Ball, Vitali Milman